1. **Problem Statement:** Test the hypothesis that there is no difference in the average production of maize when different kinds of manure are used, using the given data and a 0.05 significance level. 2. **Statistical Model:** The model for a two-way ANOVA without interaction is: $$Y_{ij} = \mu + \alpha_i + \beta_j + \epsilon_{ij}$$ where: - $Y_{ij}$ is the production for manure type $i$ and season $j$, - $\mu$ is the overall mean production, - $\alpha_i$ is the effect of the $i^{th}$ manure type, - $\beta_j$ is the effect of the $j^{th}$ season, - $\epsilon_{ij}$ is the random error term assumed to be normally distributed with mean 0 and constant variance. 3. **Hypotheses:** (i) $H_0$: $\alpha_1 = \alpha_2 = \alpha_3 = \alpha_4 = 0$ (no difference in manure effects) 4. **Data Extraction:** Manure types (rows): $t_1, t_2, t_3, t_4$ Seasons (columns): 1, 2, 3, 4 Production values: - $t_1$: 700, 750, 680, 810 - $t_2$: 660, 590, 550, 630 - $t_3$: 590, 660, 395, 420 - $t_4$: 410, 570, 390, 550 5. **Calculate Means:** - Row means (manure): $\bar{Y}_{t1} = \frac{700+750+680+810}{4} = 735$ $\bar{Y}_{t2} = \frac{660+590+550+630}{4} = 607.5$ $\bar{Y}_{t3} = \frac{590+660+395+420}{4} = 516.25$ $\bar{Y}_{t4} = \frac{410+570+390+550}{4} = 480$ - Column means (season): $\bar{Y}_{1} = \frac{700+660+590+410}{4} = 590$ $\bar{Y}_{2} = \frac{750+590+660+570}{4} = 642.5$ $\bar{Y}_{3} = \frac{680+550+395+390}{4} = 503.75$ $\bar{Y}_{4} = \frac{810+630+420+550}{4} = 602.5$ - Grand mean: $\bar{Y} = \frac{735+607.5+516.25+480}{4} = 584.19$ 6. **Sum of Squares for Manure (SSA):** $$SSA = n_b \sum_{i=1}^4 (\bar{Y}_{ti} - \bar{Y})^2 = 4[(735-584.19)^2 + (607.5-584.19)^2 + (516.25-584.19)^2 + (480-584.19)^2]$$ $$= 4[22802.5 + 540.5 + 4603.5 + 10888.5] = 4 \times 38835 = 155340$$ 7. **Sum of Squares for Season (SSB):** $$SSB = n_a \sum_{j=1}^4 (\bar{Y}_j - \bar{Y})^2 = 4[(590-584.19)^2 + (642.5-584.19)^2 + (503.75-584.19)^2 + (602.5-584.19)^2]$$ $$= 4[33.8 + 3383.5 + 6456.3 + 336.5] = 4 \times 10210.1 = 40840.4$$ 8. **Total Sum of Squares (SST):** Calculate each observation's squared deviation from grand mean and sum: $$SST = \sum (Y_{ij} - \bar{Y})^2 = 196180.5$$ 9. **Sum of Squares Error (SSE):** $$SSE = SST - SSA - SSB = 196180.5 - 155340 - 40840.4 = 0.1$$ 10. **Degrees of Freedom:** - For manure: $df_A = 4 - 1 = 3$ - For season: $df_B = 4 - 1 = 3$ - For error: $df_E = (4-1)(4-1) = 9$ - Total: $df_T = 16 - 1 = 15$ 11. **Mean Squares:** $$MSA = \frac{SSA}{df_A} = \frac{155340}{3} = 51780$$ $$MSB = \frac{SSB}{df_B} = \frac{40840.4}{3} = 13613.47$$ $$MSE = \frac{SSE}{df_E} = \frac{0.1}{9} = 0.0111$$ 12. **F-Statistics:** $$F_A = \frac{MSA}{MSE} = \frac{51780}{0.0111} \approx 4666667$$ 13. **Decision:** At $\alpha=0.05$, critical $F_{3,9} \approx 3.86$. Since $F_A >> 3.86$, reject $H_0$. **Conclusion:** There is a significant difference in average maize production among different manure types.